Beside their interesting electronic transport properties, due to
very weak spin orbit interaction [1]}, which leads to a
long spin flip length (\(\sim 1 \mu m\) ) [2]}, the graphene -
based structures also offer a good potential for spin-polarized
electronics. Actually, graphene is not a natural ferromagnet.
However, recent works have shown that ferromagnetism and
spin-polarized states can be introduced in graphene, e. g., by
doping and by defect [3]}, [4]}, [5]} or by applying an
external electric field [6]}. Especially, Haugen et. al.
[7]} suggested that the ferromagnetic correlations can be
created in graphene by the so-called proximity effect. The exchange
splitting induced by depositing a ferromagnetic insulator \(EuO\)  on
the graphene sheet was then roughly estimated to be about 5 meV.
The effect has been also experimentally demonstrated
[2]}, [9]}. Motivated by these features, some other works
have predicted and discussed the controllability of spin current by
a ferromagnetic gate in \(2D\) -graphene structures
[10]}, [11]}, [12]}. The spin current was found to be an
oscillatory function of the potential barrier, which can be tuned by
the gate voltage, and its amplitude is never damped by the increase
of the width and the height of barrier [10]}. However, the
spin polarization is not so high, e. g., its maximum value is just
about \(30\%\) . In addition, the other spin-dependent properties of
graphene such as spin field effect transistor [14]}, spin
Hall effects [1]}, [16]}, spin valve effects
[17]}, [18]}, [19]}, [20]} have been also investigated
extensively. Especially, the giant magneto-resistance has been
explored theoretically and discussed in the structures of armchair
[20]} and zigzag [19]} GNRs connecting two
conducting electrodes. The authors predicted that it can reach the
high value of \(100\%\)  [19]}.
